Sunday, January 10, 2010


Innumeracy is "the mathematical counterpart of illiteracy," says computer scientist Douglas Hofstadter. It describes "a person’s inability to make sense of the numbers that run their lives." Although Dr. Hofstadter coined the term, mathematician John Allen Paulos made the word part of the common vocabulary with his book, Innumeracy: Mathematical Illiteracy and Its Consequences. Since its publication in 1989, it has been translated into thirteen languages.

Dr. Paulos is not saying that everyone needs to be a mathematician, but rather one must be able to understand the consequences in everyday life of how our society uses numbers. The consequences of innumeracy are not as visible or obvious as the consequences of illiteracy. Nonetheless, they can be just as profound when they contribute to "confused personal decisions, muddled governmental policies, and an increased susceptibility to pseudoscience." He wonders if this is why people would buy a home having a $2,000 monthly mortgage when they only make $4,000 a month? Or why a person might borrow money at thirty percent interest in order to buy a large screen plasma television?

The difference between illiteracy and innumeracy is that the former affects mostly uneducated people whereas the latter affects people who are both intelligent and well-educated Many people who are extremely competent in many fields of knowledge and can understand complicated issues may tremble when confronted with numbers or probability.

For example, some people won’t travel to other countries because they are afraid they might be killed by a terrorist attack. Yet, they continue to travel locally in their car. The odds? Your chances of dying overseas by a terrorist attack is about 1.6 chances out of a million, yet you have a 1 chance in 5,300 of dying in the family car.

Journalist, Chip Scanlan, tells about his innumeracy and what negative effects it had in his professional career as a writer. "In high school and college, I was a terrible math student. I flunked geometry, was totally bewildered by algebra. Trigonometry I ran from. I had trouble balancing a checkbook. As a reporter, I was painfully aware of my innumeracy every time a percentage appeared in my story. Budget stories made me cringe. Without math skills I was not as effective a journalist, and my readers weren't as well-served as they could have been. It wasn't just the mistakes I made or the agonies I went through trying to figure things out. As a reporter I regurgitated statistics without understanding them because I didn't feel capable of interpreting them. I'm sure I missed stories and screwed up others because of my weak math skills." He goes on to say his dilemma is not uncommon in the journalism business.

Sometimes, people can make numerical claims that are publicly embarrassing. Years ago when the speed limit debate was taking place in America, a Washington lobbyist, Clarence Ditlow, made a public, numerical goof. He published an article explaining why he believed raising the speed limit above 55 miles per hour would be dangerous.

His heart was in the right place because he was trying to educate people against driving too fast by explaining how far a car can travel in one second. He made the following calculation: "A vehicle traveling 55 mph covers 807 feet in one second, almost three football fields. At 70 mph, a vehicle covers 220 feet more or 1027 feet in one second." Anyone who has driven a car knows they are not going the length of three football fields each second. If you care to do the calculations, you will find that at seventy miles an hour you only travel 103 feet each second.

Innumeracy has many faces for many people. One such face is seen when people say that "there is no such thing as a coincidence" — a belief that philosopher Sigmund Freud held dearly to his heart. By underestimating the likelihood of coincidences, people believe that something strange or mystical must be afoot when a coincidence happens. For example, most people would consider it a coincidence that if in a room of only twenty-three people any two people would have the same birthday. "Common sense math" appears to support this notion. Most people would calculate it this way. Since we would need 367 (counting a leap year) people in a room to be absolutely certain of two people sharing the same birthday, then it seems "obvious" that there is only a 6% change of this happening with only 23 people in a room (23 divided by 367).

Innumeracy can also raise its head in financial matters. Assume your friend invested $1000 in the stock market and the next day her investment went up 60%. Yay! The next day the price dropped 40%. Your friend might shrug and say "Great, I’m still 20% ahead in my investment." "Wrong," you would say. You would tell her that she actually lost $40 of her original investment. Here’s how you would explain it to her. Her 60% gain the first day would leave her with $1600. The next day when this $1600 went down by 40% she was left with only $960. If this doesn’t make sense you can easily check it with a simple calculator.

Statistics is one of the fastest ways for innumeracy to come bubbling to the top in any conversation (if no mathematicians are present). Although developed in the eighteenth century mainly for the purpose of developing a theory of gambling games, statistics today are the mainstay of pollsters. Commonly, innumerate people decry the results of polls because they don’t understand how it is done. The statistics that comprise polling can be baffling because it seems impossible that enough people could be polled to account for a country’s entire population. How can asking a small group of people what they think of an issue be used to say that a certain percentage of all Americans believe something specific about that issue?

Additionally, we know that people are not honest all the time so how do we know that people being asked personal questions by pollsters are going to be honest? Mathematicians have come up with a clever solution to the problem of lying that could be used by pollsters to account for any lying from the people they question. Dr. Paulos gives an example in his book of how someone could estimate the number of people in this country who might engage in a potentially embarrassing sex act (use your imagination here). Suppose you randomly asked 1000 people whether or not they engaged in this unusual sex act. Just asking the question is useless because there obvious reasons why a person would avoid being honest about their sexual behavior.

However, let’s say that each person is asked to flip a coin of their choosing. Before the question is asked they are told to flip the coin and privately look at the result of the coin flip. If the coin lands on heads, they are supposed to answer "yes" to the question being asked even if they have to lie. If the coin shows tails, they are to answer truthfully. Now the person being polled knows that nobody will ever be able to know if he or she answered the question honestly or not.

Next, each person flips their coin and looks at it in secret prior to being asked the sex act question. Some say "yes ," others say "no." When we have finished asking the thousand people let’s say we find that 620 of them answered yes. So, how do we interpret the results? Simple. We know that 500 people answered yes because odds of getting heads is 50%. We throw out these answers. Now we know that of the 500 people left who answered truthfully, 120 people, or twenty-four percent, answered "yes"honestly about their sexual behavior.

One of the ways that statistics teachers teach numeracy in an introductory statistic course is to have their students solve the well-known Birthday Problem mentioned above. With simple math tools of probability, it can easily be shown that there is a 50% chance that a room of 23 people will have the same birthday. Gamblers have used this fact as a "bar bet" for years by betting someone with odds in their favor. The gambler would rig the odds so that even though they lost half the time, they would win more money when they won the bet than they would lose when they lost the bet. There are many variations of this non-obvious observation. Dr. Paulos notices that "at least two people living in Philadelphia must have the same number of hairs on their heads." Figure that one out.

There is an interesting stock market scam that is based on innumeracy but isn't legal so you may not run into it. It goes something like this. Let’s say you received an unsolicited email from a person claiming to be a financial advisor and he was predicting the direction a particular stock would take in the following week. The week goes by and the prediction comes true. Your next email explains his company can do this because of their just-developed proprietary computer model. This email also makes another prediction of the same stock. Surprisingly, they are also correct on this prediction. This happens six weeks in a row and by now you are becoming interesting in how they can do this and seeing mega-dollar signs in your eyes. You are hooked because now the "advisor" tells you that for a reasonable sum of money you can buy into their program and become wealthy in a matter of months. Most people would jump at the chance.

Here’s the problem. What you would not know is that this person sent out the same first letter to 32,000 people. Half of these letters told people the stock would rise, the other half are told the stock would drop. This means that 16,000 people would have gotten an email with the correct prediction. Which ever way the stock actual goes, these 16,000 people who got the correct prediction would get a second letter with a new prediction. The "advisor" would use the same principle: 8,000 people would be told the stock would go up while the other half that it would go down. This pattern continues until 500 people have been given six correct "predictions" in a row for the stock.

These are the "marks" for the scamer. They are all now sent the "if you pay a fee you will have untold wealth" letter. They are told that they only have to pay $500 in order to win big next week. They each figure they can easily get their $500 back in just a week. Since you don’t know 499 other people just got the same letter you send in your money and start planning for retirement. Since everyone sends in the money, the stock "advisor" is now $25,000 richer in just six weeks. Of course, nobody ever hears from this miraculous advisor again.

Belief in predictive dreaming and precognition is also enhanced through innumeracy. How many people have you heard say something like this? "I dreamed my brother, who lived in another country, died one night and when I awoke I received a phone call saying he had had a heart attack in the night and died." This experience can feel spooky to both the dreamer and people who hear the story.

Simple math can easily explain this precognition. I won’t bore you with the calculations but the numbers show that of all the people in the world who remember their dreams (we usually have more dreams than we can remember) about 3.6 percent of all the people will have a predictive dream on any given night. Although that number seems small it is significant because there are so many people in the world. We can expect that millions of people would have at least one precognitive dream every year while hundreds of thousands would do so each night.

When innumeracy and human emotions are combined (which invariably they are) we find that most people make many decisions that are wrong -- especially when it involves risk. It has been shown many times that people will avoid risks when seeking gains, but choose risks to avoid losses. This principle was co-discovered by Daniel Kahneman, a psychologist and Nobel prize winner in economics.

Let’s say you had the choice of winning $30,000 or making a bet that has an 80% chance of of winning $40,000 or a 20% of winning nothing. Most people would take the thirty grand even though the average gain from the second choice would be $32,000.

If the scenario were changed so that you it was certain you would lose $30,000 or given a second choice where there was an 80% of losing $40,000 or a 20% chance of losing nothing. In this case, most people would take the second choice even though it would be wiser to take the 80% change of losing $40,000.

Numbers are a part of who we are because we live in a world saturated by numbers. Technology, which is all about numbers, is advancing exponentially as is information from our technology. Thanks to the ubiquity of computers and the Internet, we are bombarded hourly with information and we have to decide whether to accept new information or reject it. Fortunately, we don’t have to make all the choices by ourselves.

We can make wise choices by using numerate people who understand numbers and know how to do legitimate research. One such example is the team of Barbara and David Mikkelson. They are excellent researchers and are experts at sniffing out rumors, urban legends and new myths. Their website, Snopes, often uses mathematics to find out the truth of dubious claims wending their way through cyberspace.

Numeracy is also required of all scientists since mathematics is the language of science. We need to listen to scientists for getting through the daily barrage of fluff, lies, and half truths poring forth from media outlets and their so-called experts. Even well-meaning and smart people can be propagators of nonsense by being innumerate. Scientists, mathematicians and engineers can help us get clarity on many issues. For example, Peter Aczel, is an audio engineer who has debunked nonsense in the world of high-end audio. His online magazine, The Audio Critic, is the place to head if you plan to spend any significant amount of money on new or even used high-end audio equipment. His back issues begin in 1991 (even though he began his magazine long before then) and well worth the read. His no-nonsense intelligence and sharp wit that does not suffer fools gladly will keep you entertained for hours.

Religion and innumeracy are common bed fellows. America is the most religious country in the world, yet for all its popularity, religion can promote certain myths as absolute truth. For instance, some religious people believe in a literal Flood as described in the book of Genesis. Fortunately, most religious people understand this story as a metaphor with much deeper meaning than can be had from a merely literal rendition.

Why is it difficult to accept this story literally? That is because the story is loaded with numbers — numbers that some people don’t take seriously or avoid even if they take the story seriously. The writer of this Genesis story says that ". . . all the hills that were under the whole heaven were covered . . ." Taken literally, this means the entire surface of the earth was covered by at least 20,000 feet of water which would make the water about four miles deep (the deepest part of our planet’s oceans in the Mariana Trench which is over 6.8 miles deep). That was a deep flood. Even worse, the Flood Story tells us that it rained steadily for 40 days and nights for a total of 960 hours. Doing the math tells us that the rainfall was at least 15 feet per hour! This is enough of a downpour to sink an aircraft carrier, let alone a small wooden ark with thousands of animals on board.

Mathematician John Allen Paulos wrote his book on innumeracy because, in his words, "I'm distressed by a society which depends so completely on mathematics and science and yet seems so indifferent to the innumeracy and scientific illiteracy of so many of its citizens."


Best, J. (2001). Damned Lies and Statistics: Untangling numbers from the media, policians, and activists. Berkeley, CA: University of California Press.

Paulos, J. (2001). Innumeracy: Mathematical illiteracy and its consequences. New York: Hill and Wang.

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